A326878 Number of topologies whose points are a subset of {1..n}.

1, 2, 7, 45, 500, 9053, 257151, 11161244, 725343385, 69407094565, 9639771895398, 1919182252611715, 541764452276876719, 214777343584048313318, 118575323291814379721651, 90492591258634595795504697, 94844885130660856889237907260, 135738086271526574073701454370969, 263921383510041055422284977248713291

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Aidar Dulliev and Daniil Naumikhin, Binomial Transform of Sequences Counting N-ary Convexities, arXiv:2606.11252 [math.GM], 2026. See p. 6 (Table 2).

Wikipedia Topological space.

Formula

From Geoffrey Critzer, Jul 12 2022: (Start)

E.g.f.: exp(x)*A(exp(x)-1) where A(x) is the e.g.f. for A001035.

a(n) = Sum_{k=0..n} binomial(n,k)*A000798(k). (End)

Example

The a(0) = 1 through a(2) = 7 topologies:
  {{}}  {{}}      {{}}
        {{},{1}}  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n]]], MemberQ[#, {}]&&SubsetQ[#, Union[Union@@@Tuples[#, 2], Intersection@@@Tuples[#, 2]]]&]], {n, 0, 4}]
(* Alternative: *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]];
a[n_] := Sum[Binomial[n, k]*A000798[[k+1]], {k, 0, n}];
a /@ Range[0, Length[A000798]-1] (* Jean-François Alcover, Dec 30 2019 *)

Crossrefs

Binomial transform of A000798 (the covering case).

Cf. A001035, A001930, A003465, A014466, A102896, A102897, A306445, A326866, A326876.

Sequence in context: A098637 A162045 A383337 * A153549 A348880 A079491

Adjacent sequences: A326875 A326876 A326877 * A326879 A326880 A326881

Keyword

nonn

Author

Gus Wiseman, Jul 30 2019

Status

approved